Using Sketchpad - Conics
Newton's Organic Construction
Isaac Newton published the following construction for a general conic section, which he called his "organic" construction. Newton's original description is almost incomprehensible. A somewhat more understandable, although still a bit dated, one is given by Salmon:
Two angles of constant magnitude move about fixed points P, Q; the intersection of two of their sides traverses a right line AA'; then the locus of V, the intersection of the other two sides will be a conic passing through P, Q.
This construction is illustrated in the sketch.
Start with a line AA' and two points (P and Q). For each point (M) on the line, draw the lines l1 = MA and l2 = MB. Rotate l1 through a fixed angle (50 degrees in this example, this is one of the constant angles described above) to get line l1'. Rotate l2 through another fixed angle (25 degrees in this example, this is the other constant angle) to get the line l2'. Look at the point V at the intersection of l1' and l2'. As M sweeps across AA', V sweeps out a conic that goes through P and Q.
You can experiment by moving P and Q around to see if you can generate other conics. With a little bit more work you can change the angles MPV and MQV.
The MacLaurin/Braikenridge construction of a general conic
Braikenridge published the following construction:
The conic is the locus of the vertex V of a triangle whose sides pass through the points A, B, C and whose base angles move on the fixed lines OX and OY.
Coolidge describes some of the controversy around who exactly should get credit for the construction. This construction is illustrated in the sketch.
Start with two lines (OX and OY, intersecting at O) and three point A, B and C. For each point M on OX, construct the line MA intersecting OY at D, the line CD and the line MC. Let V be the intersection of MC and DB. Then as M sweeps across the line OX, the point V sweeps out a conic that goes through B, C and E. The triangle mentioned in the original description is MDV.
Although this construction was published significantly before the development of the "projective" school, it is not too hard to see that teh pencil of lines through C is projectively related to the pencil of lines through B (by perspectivities on OX and OY) so the intersections of corresponding lines (the points V) from each pencil lie on a conic. See Salmon.
Try moving the points A, B and/or C to generate different conics.
Coolidge, Julian Lowell, A history of the conic sections and quadric surfaces, The Clarendon Press, Oxford, 1945.
Salmon, George, A treatise on conic sections: containing an account of some of the more important modern algebraic and geometric methods, Hodges and Smith, Dublin 1850. (I don't think this is the first edition of this classic, amd it certainly isn't the last!)
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